If the height
of a triangle is five inches less than the length of its base, and if
the area of the triangle is 52
square inches, find the base and the height.

They have given me a
relationship between the height and the base, and have given me the
value of the area. So I'll need to use the formula for the area of a
triangle with a given base and height, and I'll need to create an expression
or equation relating the height and base.

The area of a triangle
is given by:

A
= ( 1/2 )bh

...where "b"
is the base and "h"
is the height (or "altitude"). I am given that the height
is five less than the base, so the equation for their relationship is:

h
= b – 5

Since I am given that
the area is 52
square inches, I can then plug the base variable, the height expression,
and the area value into the formula for the area of a triangle, and
see where this leads:

I can safely ignore the
extraneous negative solution. (A solution which is "extraneous",
pronounced "ek-STRAY-nee-uss", is a number that is a valid
solution to the equation, but is not a relevant value in the context
of the word problem. In this case, lengths cannot be negative.) This
means that b
= 13, so h
= b – 5 = 13 – 5 = 8.

The base is 13
inches, and the height is 8
inches.

In the last exercise above,
I solved for one value (the length of the base) and then back-solved for
the other value (the length of the height). This other value turned out
to be the same as the extraneous value, except
for the sign change.
Warning: Do not assume that you can get both of your answers by arbitrarily
changing the sign on the extraneous solution. This does not always work,
it is mathematically wrong, it annoys your teacher, and it can get you
in trouble further down the line.

Another triangle formula
you should remember is the Pythagorean Theorem:

Take a right-angled triangle,
and square the lengths of all three sides. If you add up the squares
of the two shorter sides, this sum will be the same value as the value
of the square of the longest side."

As a formula, the Pythagorean
Theorem is often stated in the form "a2
+ b2 = c2",
where a and
b are
the lengths of the two legs (the two shorter sides) and c is
the length of the hypotenuse (being the longest side, opposite the right
angle).

ADVERTISEMENT

If the sum of the sides
of a right triangle is 49
inches and the hypoteneuse is 41
inches, find the two sides.

In this case, either
solution will do. If b
= 9, then a
= 49 – b = 49 – 9 = 40. Or
if b
= 40, then a
= 49 – b = 49 – 40 = 9. Since
the problem didn't specify which of the two legs is longer, it doesn't
matter which one I call "a"
and which one I call "b".
The answer is:

One side is forty
inches long, and the other side is nine inches long.

A wood frame for pouring
concrete has an interior perimeter of 14 meters. Its length is one meter
greater than its width. The frame is to be braced with twelve-gauge
steel cross-wires. Assuming an extra half-meter of wire is used at either
end of a cross-wire for anchoring, what length of wire should be cut
for each brace?

I don't care that the
wire is steel; I don't care that they're pouring concrete into a wood
frame. All I need is the geometrical information: this is a rectangle
with a certain perimeter and a certain relationship between the length
and the width. They're asking me, effectively, to find the length of
the diagonal. And this diagonal, together with the length and the width,
will form a right triangle. So the perimeter formula for a rectangle
may be useful, as may the Pythagorean Theorem.

width: wlength:
w
+ 1perimeter
formula: 14
= 2(w + 1) + 2(w)

14 = 2w
+ 2 + 2w14
= 4w + 212
= 4w3 =
w

Then the length, being
one unit larger, is 4,
and the Pythagorean Theorem lets me find the length of the diagonal
d:

32
+ 42 = d29 +
16 = 25 = d25 =
d

Adding a half-meter at
either end of the wire, I find that:

each wire should
be cut to a length of six meters

Another useful triangle
fact is that the measures of any triangle's three angles add up to 180
degrees.

The smallest angle
of a triangle is two-thirds the size of the middle angle, and the middle
angle is three-sevenths of the largest angle. Find all three angle measures.

The smallest angle is
defined in terms of the middle angle, but the middle angle is defined
in terms of the largest angle. So it makes most sense to pick a variable
for the measure of the largest angle, and then create expressions for
the middle and then the smallest angles, using that variable.

I'll let "ß"
stand for "beta", the largest angle, or, rather, for the measure
of the largest angle. Then the middle angle has a measure of (
3/7 )ß.
The smallest angle is two-thirds of the middle angle, so it has a measure
of ( 2/3
)( 3/7 )ß = ( 2/7
)ß. Then my
angle-sum formula is:

ß
+ ( 3/7 )ß + ( 2/7
)ß = 1807ß
+ 3ß + 2ß = 126012ß
= 1260ß
= 105

So the largest angle
has a measure of 105
degrees. The middle angle is then: